Battery powered equipment typically does not respond well to over-voltage. I've used battery powered radio and TV gear for 30 years, and I've always had the best results using batteries of the specific voltage required by the device. I also am somewhat nervous about the potential for damaging a device if I subject it to over-voltage.
Wikipedia shows specs for voltage in 9V batteries, and it appears that 9.6V is a normal initial voltage for lithium polymer rechargeables and some NIMH batteries. However, this is .6V higher than the alkaline batteries for which the device was most likely designed.
http://en.wikipedia.org/wiki/Nine-volt_battery
It does mention this: "The Lithium PP3 (U9VL-J, U9VL-X, U9VL-FP, U9VL-FP, Li9V & ER9V), is a consumer-replaceable, high energy density battery that can last up to 5 times longer than ordinary alkaline 9-volt batteries and up to 10 times longer than carbon-zinc batteries in many applications. An additional property of the lithium PP3 is its long shelf life, up to 10-years for the aluminum-jacketed U9VL-J battery."
Where you CAN cheat is on the amperage. A battery with more milliamp-hours (mAh) will last longer under use. Here's a reference chart that pre-dates the latest battery technologies:
http://www.techlib.com/reference/batteries.html
it also tells you how to calculate the length of the battery life based on how much current your device draws.
"To determine the battery life, divide the capacity by the actual load current to get the hours of life. A circuit that draws 10 ma powered by a 9 volt rectangular battery will operate about 50 hours: 500 mAh /10 mA = 50 hours."
There are also differences in the way voltage drops off under load, depending on whether you are using a rechargeable or alkaline battery. Ni-Cad batteries have some very bad habits in this respect - they can hit a "step" where the voltage drops suddenly to a lower level. The battery still can function, but this drop and fools the device into thinking the battery is discharged and it stops working. NIMH batteries normally don't have this problem.
On a practical basis, I stick with conventional alkaline 9V batteries because they're dependable, inexpensive and aren't terrible in landfills. I don't trust rechargeable batteries unless I take them right off the charger and drop them into the device. I've had too many problems with voltage based on the number of charge cycles, etc.